Tangential Tallini sets in finite Grassmannians of lines

A Tallini set in a semilinear space is a set B of points, such that each line not contained in B intersects B in at most two points. In this paper, the following notion of a tangential Tallini set in the Grassmannian Γn,1,q, q odd, is investigated: a Tallini set is called tangential when it meets every ruled plane (i.e. the set of lines contained in a plane of PG(n, q)) in either q + 1 or q2 + q + 1 elements. A Tallini set QB in PG(n, q) can be associated with each tangential Tallini set B in Γn,1,q. Each l∈B is a line of PG(n, q) intersecting QB in either 0, or 1, or q + 1 points; when n ≠ 4 and B is covered by (n - 2)-dimensional projective subspaces of γn,1,q the first case does not occur. If B is a tangential Tallini set in γn,1,q covered by (n - 2)-dimensional subspaces, any of which is in PG(n, q) the set of all lines through a point and in a hyperplane, then either QB is a quadric, and B is the set of all lines contained in, or tangent to, QB, or B is a linear complex.